P. 148 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14701-14800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mplelf 14701*	A polynomial is defined as a function on the coefficients. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- P = ( I mPoly R )   &    |- K = ( Base ` R )   &    |- B = ( Base ` P )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- ( ph -> X e. B )   =>    |- ( ph -> X : D --> K )
Theorem	mplsubgfilemm 14702*	Lemma for mplsubgfi 14705. There exists a polynomial. (Contributed by Jim Kingdon, 21-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> E. j j e. U )
Theorem	mplsubgfilemcl 14703	Lemma for mplsubgfi 14705. The sum of two polynomials is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- .+ = ( +g ` S )   =>    |- ( ph -> ( X .+ Y ) e. U )
Theorem	mplsubgfileminv 14704	Lemma for mplsubgfi 14705. The additive inverse of a polynomial is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. U )   &    |- N = ( invg ` S )   =>    |- ( ph -> ( N ` X ) e. U )
Theorem	mplsubgfi 14705	The set of polynomials is closed under addition, i.e. it is a subgroup of the set of power series. (Contributed by Mario Carneiro, 8-Jan-2015.) (Proof shortened by AV, 16-Jul-2019.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> U e. (SubGrp ` S ) )
Theorem	mpl0fi 14706*	The zero polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
|- P = ( I mPoly R )   &    |- O = ( 0g ` R )   &    |- .0. = ( 0g ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> .0. = ( x e. ( NN0 ^m I ) |-> O ) )
Theorem	mplplusgg 14707	Value of addition in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- Y = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- .+ = ( +g ` Y )   =>    |- ( ( I e. V /\ R e. W ) -> .+ = ( +g ` S ) )
Theorem	mpladd 14708	The addition operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- P = ( I mPoly R )   &    |- B = ( Base ` P )   &    |- .+ = ( +g ` R )   &    |- .+b = ( +g ` P )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .+b Y ) = ( X oF .+ Y ) )
Theorem	mplnegfi 14709	The negative function on multivariate polynomials. (Contributed by SN, 25-May-2024.)
|- P = ( I mPoly R )   &    |- B = ( Base ` P )   &    |- N = ( invg ` R )   &    |- M = ( invg ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( M ` X ) = ( N o. X ) )
Theorem	mplgrpfi 14710	The polynomial ring is a group. (Contributed by Mario Carneiro, 9-Jan-2015.)
|- P = ( I mPoly R )   =>    |- ( ( I e. Fin /\ R e. Grp ) -> P e. Grp )
PART 9  BASIC TOPOLOGY
9.1  Topology
9.1.1  Topological spaces A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union), and it may sometimes be more convenient to consider topologies without reference to the underlying set.
9.1.1.1  Topologies
Syntax	ctop 14711	Syntax for the class of topologies.
class Top
Definition	df-top 14712*	Define the class of topologies. It is a proper class. See istopg 14713 and istopfin 14714 for the corresponding characterizations, using respectively binary intersections like in this definition and nonempty finite intersections. The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241. (Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)
|- Top = { x | ( A. y e. ~P x U. y e. x /\ A. y e. x A. z e. x ( y i^i z ) e. x ) }
Theorem	istopg 14713*	Express the predicate " J is a topology". See istopfin 14714 for another characterization using nonempty finite intersections instead of binary intersections. Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use T to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- ( J e. A -> ( J e. Top <-> ( A. x ( x C_ J -> U. x e. J ) /\ A. x e. J A. y e. J ( x i^i y ) e. J ) ) )
Theorem	istopfin 14714*	Express the predicate " J is a topology" using nonempty finite intersections instead of binary intersections as in istopg 14713. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.)
|- ( J e. Top -> ( A. x ( x C_ J -> U. x e. J ) /\ A. x ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) -> |^| x e. J ) ) )
Theorem	uniopn 14715	The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
|- ( ( J e. Top /\ A C_ J ) -> U. A e. J )
Theorem	iunopn 14716*	The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
|- ( ( J e. Top /\ A. x e. A B e. J ) -> U_ x e. A B e. J )
Theorem	inopn 14717	The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
|- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A i^i B ) e. J )
Theorem	fiinopn 14718	The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
|- ( J e. Top -> ( ( A C_ J /\ A =/= (/) /\ A e. Fin ) -> |^| A e. J ) )
Theorem	unopn 14719	The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A u. B ) e. J )
Theorem	0opn 14720	The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
|- ( J e. Top -> (/) e. J )
Theorem	0ntop 14721	The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
|- -. (/) e. Top
Theorem	topopn 14722	The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
|- X = U. J   =>    |- ( J e. Top -> X e. J )
Theorem	eltopss 14723	A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
|- X = U. J   =>    |- ( ( J e. Top /\ A e. J ) -> A C_ X )
9.1.1.2  Topologies on sets
Syntax	ctopon 14724	Syntax for the function of topologies on sets.
class TopOn
Definition	df-topon 14725*	Define the function that associates with a set the set of topologies on it. (Contributed by Stefan O'Rear, 31-Jan-2015.)
|- TopOn = ( b e. _V |-> { j e. Top | b = U. j } )
Theorem	funtopon 14726	The class TopOn is a function. (Contributed by BJ, 29-Apr-2021.)
|- Fun TopOn
Theorem	istopon 14727	Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- ( J e. (TopOn ` B ) <-> ( J e. Top /\ B = U. J ) )
Theorem	topontop 14728	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( J e. (TopOn ` B ) -> J e. Top )
Theorem	toponuni 14729	The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( J e. (TopOn ` B ) -> B = U. J )
Theorem	topontopi 14730	A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- J e. (TopOn ` B )   =>    |- J e. Top
Theorem	toponunii 14731	The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- J e. (TopOn ` B )   =>    |- B = U. J
Theorem	toptopon 14732	Alternative definition of Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- X = U. J   =>    |- ( J e. Top <-> J e. (TopOn ` X ) )
Theorem	toptopon2 14733	A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
|- ( J e. Top <-> J e. (TopOn ` U. J ) )
Theorem	topontopon 14734	A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.)
|- ( J e. (TopOn ` X ) -> J e. (TopOn ` U. J ) )
Theorem	toponrestid 14735	Given a topology on a set, restricting it to that same set has no effect. (Contributed by Jim Kingdon, 6-Jul-2022.)
|- A e. (TopOn ` B )   =>    |- A = ( A ↾t B )
Theorem	toponsspwpwg 14736	The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.)
|- ( A e. V -> (TopOn ` A ) C_ ~P ~P A )
Theorem	dmtopon 14737	The domain of TopOn is _V. (Contributed by BJ, 29-Apr-2021.)
|- dom TopOn = _V
Theorem	fntopon 14738	The class TopOn is a function with domain _V. (Contributed by BJ, 29-Apr-2021.)
|- TopOn Fn _V
Theorem	toponmax 14739	The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
|- ( J e. (TopOn ` B ) -> B e. J )
Theorem	toponss 14740	A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
|- ( ( J e. (TopOn ` X ) /\ A e. J ) -> A C_ X )
Theorem	toponcom 14741	If K is a topology on the base set of topology J, then J is a topology on the base of K. (Contributed by Mario Carneiro, 22-Aug-2015.)
|- ( ( J e. Top /\ K e. (TopOn ` U. J ) ) -> J e. (TopOn ` U. K ) )
Theorem	toponcomb 14742	Biconditional form of toponcom 14741. (Contributed by BJ, 5-Dec-2021.)
|- ( ( J e. Top /\ K e. Top ) -> ( J e. (TopOn ` U. K ) <-> K e. (TopOn ` U. J ) ) )
Theorem	topgele 14743	The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
|- ( J e. (TopOn ` X ) -> ( { (/) , X } C_ J /\ J C_ ~P X ) )
9.1.1.3  Topological spaces
Syntax	ctps 14744	Syntax for the class of topological spaces.
class TopSp
Definition	df-topsp 14745	Define the class of topological spaces (as extensible structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
|- TopSp = { f | ( TopOpen ` f ) e. (TopOn ` ( Base ` f ) ) }
Theorem	istps 14746	Express the predicate "is a topological space". (Contributed by Mario Carneiro, 13-Aug-2015.)
|- A = ( Base ` K )   &    |- J = ( TopOpen ` K )   =>    |- ( K e. TopSp <-> J e. (TopOn ` A ) )
Theorem	istps2 14747	Express the predicate "is a topological space". (Contributed by NM, 20-Oct-2012.)
|- A = ( Base ` K )   &    |- J = ( TopOpen ` K )   =>    |- ( K e. TopSp <-> ( J e. Top /\ A = U. J ) )
Theorem	tpsuni 14748	The base set of a topological space. (Contributed by FL, 27-Jun-2014.)
|- A = ( Base ` K )   &    |- J = ( TopOpen ` K )   =>    |- ( K e. TopSp -> A = U. J )
Theorem	tpstop 14749	The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)
|- J = ( TopOpen ` K )   =>    |- ( K e. TopSp -> J e. Top )
Theorem	tpspropd 14750	A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- ( ph -> ( Base ` K ) = ( Base ` L ) )   &    |- ( ph -> ( TopOpen ` K ) = ( TopOpen ` L ) )   =>    |- ( ph -> ( K e. TopSp <-> L e. TopSp ) )
Theorem	topontopn 14751	Express the predicate "is a topological space". (Contributed by Mario Carneiro, 13-Aug-2015.)
|- A = ( Base ` K )   &    |- J = (TopSet ` K )   =>    |- ( J e. (TopOn ` A ) -> J = ( TopOpen ` K ) )
Theorem	tsettps 14752	If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.)
|- A = ( Base ` K )   &    |- J = (TopSet ` K )   =>    |- ( J e. (TopOn ` A ) -> K e. TopSp )
Theorem	istpsi 14753	Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)
|- ( Base ` K ) = A   &    |- ( TopOpen ` K ) = J   &    |- A = U. J   &    |- J e. Top   =>    |- K e. TopSp
Theorem	eltpsg 14754	Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.)
|- K = { <. ( Base ` ndx ) , A >. , <. (TopSet ` ndx ) , J >. }   =>    |- ( J e. (TopOn ` A ) -> K e. TopSp )
Theorem	eltpsi 14755	Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- K = { <. ( Base ` ndx ) , A >. , <. (TopSet ` ndx ) , J >. }   &    |- A = U. J   &    |- J e. Top   =>    |- K e. TopSp
9.1.2  Topological bases
Syntax	ctb 14756	Syntax for the class of topological bases.
class TopBases
Definition	df-bases 14757*	Define the class of topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 14759). Note that "bases" is the plural of "basis". (Contributed by NM, 17-Jul-2006.)
|- TopBases = { x | A. y e. x A. z e. x ( y i^i z ) C_ U. ( x i^i ~P ( y i^i z ) ) }
Theorem	isbasisg 14758*	Express the predicate "the set B is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
|- ( B e. C -> ( B e. TopBases <-> A. x e. B A. y e. B ( x i^i y ) C_ U. ( B i^i ~P ( x i^i y ) ) ) )
Theorem	isbasis2g 14759*	Express the predicate "the set B is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
|- ( B e. C -> ( B e. TopBases <-> A. x e. B A. y e. B A. z e. ( x i^i y ) E. w e. B ( z e. w /\ w C_ ( x i^i y ) ) ) )
Theorem	isbasis3g 14760*	Express the predicate "the set B is a basis for a topology". Definition of basis in [Munkres] p. 78. (Contributed by NM, 17-Jul-2006.)
|- ( B e. C -> ( B e. TopBases <-> ( A. x e. B x C_ U. B /\ A. x e. U. B E. y e. B x e. y /\ A. x e. B A. y e. B A. z e. ( x i^i y ) E. w e. B ( z e. w /\ w C_ ( x i^i y ) ) ) ) )
Theorem	basis1 14761	Property of a basis. (Contributed by NM, 16-Jul-2006.)
|- ( ( B e. TopBases /\ C e. B /\ D e. B ) -> ( C i^i D ) C_ U. ( B i^i ~P ( C i^i D ) ) )
Theorem	basis2 14762*	Property of a basis. (Contributed by NM, 17-Jul-2006.)
|- ( ( ( B e. TopBases /\ C e. B ) /\ ( D e. B /\ A e. ( C i^i D ) ) ) -> E. x e. B ( A e. x /\ x C_ ( C i^i D ) ) )
Theorem	fiinbas 14763*	If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( B e. C /\ A. x e. B A. y e. B ( x i^i y ) e. B ) -> B e. TopBases )
Theorem	baspartn 14764*	A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( ( P e. V /\ A. x e. P A. y e. P ( x = y \/ ( x i^i y ) = (/) ) ) -> P e. TopBases )
Theorem	tgval2 14765*	Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 14778) that ( topGen ` B ) is indeed a topology (on U. B, see unitg 14776). See also tgval 13335 and tgval3 14772. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> ( topGen ` B ) = { x | ( x C_ U. B /\ A. y e. x E. z e. B ( y e. z /\ z C_ x ) ) } )
Theorem	eltg 14766	Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> ( A e. ( topGen ` B ) <-> A C_ U. ( B i^i ~P A ) ) )
Theorem	eltg2 14767*	Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> ( A e. ( topGen ` B ) <-> ( A C_ U. B /\ A. x e. A E. y e. B ( x e. y /\ y C_ A ) ) ) )
Theorem	eltg2b 14768*	Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> ( A e. ( topGen ` B ) <-> A. x e. A E. y e. B ( x e. y /\ y C_ A ) ) )
Theorem	eltg4i 14769	An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)
|- ( A e. ( topGen ` B ) -> A = U. ( B i^i ~P A ) )
Theorem	eltg3i 14770	The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
|- ( ( B e. V /\ A C_ B ) -> U. A e. ( topGen ` B ) )
Theorem	eltg3 14771*	Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.)
|- ( B e. V -> ( A e. ( topGen ` B ) <-> E. x ( x C_ B /\ A = U. x ) ) )
Theorem	tgval3 14772*	Alternate expression for the topology generated by a basis. Lemma 2.1 of [Munkres] p. 80. See also tgval 13335 and tgval2 14765. (Contributed by NM, 17-Jul-2006.) (Revised by Mario Carneiro, 30-Aug-2015.)
|- ( B e. V -> ( topGen ` B ) = { x | E. y ( y C_ B /\ x = U. y ) } )
Theorem	tg1 14773	Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
|- ( A e. ( topGen ` B ) -> A C_ U. B )
Theorem	tg2 14774*	Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006.)
|- ( ( A e. ( topGen ` B ) /\ C e. A ) -> E. x e. B ( C e. x /\ x C_ A ) )
Theorem	bastg 14775	A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.)
|- ( B e. V -> B C_ ( topGen ` B ) )
Theorem	unitg 14776	The topology generated by a basis B is a topology on U. B. Importantly, this theorem means that we don't have to specify separately the base set for the topological space generated by a basis. In other words, any member of the class TopBases completely specifies the basis it corresponds to. (Contributed by NM, 16-Jul-2006.) (Proof shortened by OpenAI, 30-Mar-2020.)
|- ( B e. V -> U. ( topGen ` B ) = U. B )
Theorem	tgss 14777	Subset relation for generated topologies. (Contributed by NM, 7-May-2007.)
|- ( ( C e. V /\ B C_ C ) -> ( topGen ` B ) C_ ( topGen ` C ) )
Theorem	tgcl 14778	Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)
|- ( B e. TopBases -> ( topGen ` B ) e. Top )
Theorem	tgclb 14779	The property tgcl 14778 can be reversed: if the topology generated by B is actually a topology, then B must be a topological basis. This yields an alternative definition of TopBases. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- ( B e. TopBases <-> ( topGen ` B ) e. Top )
Theorem	tgtopon 14780	A basis generates a topology on U. B. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- ( B e. TopBases -> ( topGen ` B ) e. (TopOn ` U. B ) )
Theorem	topbas 14781	A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
|- ( J e. Top -> J e. TopBases )
Theorem	tgtop 14782	A topology is its own basis. (Contributed by NM, 18-Jul-2006.)
|- ( J e. Top -> ( topGen ` J ) = J )
Theorem	eltop 14783	Membership in a topology, expressed without quantifiers. (Contributed by NM, 19-Jul-2006.)
|- ( J e. Top -> ( A e. J <-> A C_ U. ( J i^i ~P A ) ) )
Theorem	eltop2 14784*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
|- ( J e. Top -> ( A e. J <-> A. x e. A E. y e. J ( x e. y /\ y C_ A ) ) )
Theorem	eltop3 14785*	Membership in a topology. (Contributed by NM, 19-Jul-2006.)
|- ( J e. Top -> ( A e. J <-> E. x ( x C_ J /\ A = U. x ) ) )
Theorem	tgdom 14786	A space has no more open sets than subsets of a basis. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 9-Apr-2015.)
|- ( B e. V -> ( topGen ` B ) ~<_ ~P B )
Theorem	tgiun 14787*	The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- ( ( B e. V /\ A. x e. A C e. B ) -> U_ x e. A C e. ( topGen ` B ) )
Theorem	tgidm 14788	The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
|- ( B e. V -> ( topGen ` ( topGen ` B ) ) = ( topGen ` B ) )
Theorem	bastop 14789	Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006.)
|- ( B e. TopBases -> ( B e. Top <-> ( topGen ` B ) = B ) )
Theorem	tgtop11 14790	The topology generation function is one-to-one when applied to completed topologies. (Contributed by NM, 18-Jul-2006.)
|- ( ( J e. Top /\ K e. Top /\ ( topGen ` J ) = ( topGen ` K ) ) -> J = K )
Theorem	en1top 14791	{ (/) } is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
|- ( J e. Top -> ( J ~~ 1o <-> J = { (/) } ) )
Theorem	tgss3 14792	A criterion for determining whether one topology is finer than another. Lemma 2.2 of [Munkres] p. 80 using abbreviations. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
|- ( ( B e. V /\ C e. W ) -> ( ( topGen ` B ) C_ ( topGen ` C ) <-> B C_ ( topGen ` C ) ) )
Theorem	tgss2 14793*	A criterion for determining whether one topology is finer than another, based on a comparison of their bases. Lemma 2.2 of [Munkres] p. 80. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
|- ( ( B e. V /\ U. B = U. C ) -> ( ( topGen ` B ) C_ ( topGen ` C ) <-> A. x e. U. B A. y e. B ( x e. y -> E. z e. C ( x e. z /\ z C_ y ) ) ) )
Theorem	basgen 14794	Given a topology J, show that a subset B satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81 using abbreviations. (Contributed by NM, 22-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.)
|- ( ( J e. Top /\ B C_ J /\ J C_ ( topGen ` B ) ) -> ( topGen ` B ) = J )
Theorem	basgen2 14795*	Given a topology J, show that a subset B satisfying the third antecedent is a basis for it. Lemma 2.3 of [Munkres] p. 81. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
|- ( ( J e. Top /\ B C_ J /\ A. x e. J A. y e. x E. z e. B ( y e. z /\ z C_ x ) ) -> ( topGen ` B ) = J )
Theorem	2basgeng 14796	Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Jim Kingdon, 5-Mar-2023.)
|- ( ( B e. V /\ B C_ C /\ C C_ ( topGen ` B ) ) -> ( topGen ` B ) = ( topGen ` C ) )
Theorem	bastop1 14797*	A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom " ( topGen ` B ) = J " to express " B is a basis for topology J " since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
|- ( ( J e. Top /\ B C_ J ) -> ( ( topGen ` B ) = J <-> A. x e. J E. y ( y C_ B /\ x = U. y ) ) )
Theorem	bastop2 14798*	A version of bastop1 14797 that doesn't have B C_ J in the antecedent. (Contributed by NM, 3-Feb-2008.)
|- ( J e. Top -> ( ( topGen ` B ) = J <-> ( B C_ J /\ A. x e. J E. y ( y C_ B /\ x = U. y ) ) ) )
9.1.3  Examples of topologies
Theorem	distop 14799	The discrete topology on a set A. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 17-Jul-2006.) (Revised by Mario Carneiro, 19-Mar-2015.)
|- ( A e. V -> ~P A e. Top )
Theorem	topnex 14800	The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4544. (Contributed by BJ, 2-May-2021.)
|- Top e/ _V